Details
| Original language | English |
|---|---|
| Pages (from-to) | 1021-1042 |
| Number of pages | 22 |
| Journal | Mathematische Annalen |
| Volume | 361 |
| Issue number | 3-4 |
| Publication status | Published - Apr 2015 |
Abstract
Let (Formula presented.) be an extension of number fields, and let (Formula presented.) be a quadratic polynomial over (Formula presented.). Let (Formula presented.) be the affine variety defined by (Formula presented.). We study the Hasse principle and weak approximation for (Formula presented.) in three cases. For (Formula presented.) and (Formula presented.) irreducible over (Formula presented.) and split in (Formula presented.), we prove the Hasse principle and weak approximation. For (Formula presented.) with arbitrary (Formula presented.), we show that the Brauer-Manin obstruction to the Hasse principle and weak approximation is the only one. For (Formula presented.) and (Formula presented.) irreducible over k, we determine the Brauer group of smooth proper models of X. In a case where it is non-trivial, we exhibit a counterexample to weak approximation.
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In: Mathematische Annalen, Vol. 361, No. 3-4, 04.2015, p. 1021-1042.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Universal torsors and values of quadratic polynomials represented by norms
AU - Derenthal, Ulrich
AU - Smeets, Arne
AU - Wei, Dasheng
N1 - Funding information: The first named author was supported by Grant DE 1646/2–1 of the Deutsche Forschungsgemeinschaft and grant 200021_124737/1 of the Schweizer Nationalfonds. The second named author was supported by a PhD fellowship of the Research Foundation—Flanders (FWO). The third named author was supported by National Key Basic Research Program of China (Grant No. 2013CB834202) and National Natural Science Foundation of China (Grant Nos. 11371210 and 11321101). This collaboration was supported by the Center for Advanced Studies of LMU München. We thank T. D. Browning, J.-L. Colliot-Thélène, C. Demarche and B. Kunyavski? for useful discussions and remarks. Finally, we thank the referee for his suggestions for improvement.
PY - 2015/4
Y1 - 2015/4
N2 - Let (Formula presented.) be an extension of number fields, and let (Formula presented.) be a quadratic polynomial over (Formula presented.). Let (Formula presented.) be the affine variety defined by (Formula presented.). We study the Hasse principle and weak approximation for (Formula presented.) in three cases. For (Formula presented.) and (Formula presented.) irreducible over (Formula presented.) and split in (Formula presented.), we prove the Hasse principle and weak approximation. For (Formula presented.) with arbitrary (Formula presented.), we show that the Brauer-Manin obstruction to the Hasse principle and weak approximation is the only one. For (Formula presented.) and (Formula presented.) irreducible over k, we determine the Brauer group of smooth proper models of X. In a case where it is non-trivial, we exhibit a counterexample to weak approximation.
AB - Let (Formula presented.) be an extension of number fields, and let (Formula presented.) be a quadratic polynomial over (Formula presented.). Let (Formula presented.) be the affine variety defined by (Formula presented.). We study the Hasse principle and weak approximation for (Formula presented.) in three cases. For (Formula presented.) and (Formula presented.) irreducible over (Formula presented.) and split in (Formula presented.), we prove the Hasse principle and weak approximation. For (Formula presented.) with arbitrary (Formula presented.), we show that the Brauer-Manin obstruction to the Hasse principle and weak approximation is the only one. For (Formula presented.) and (Formula presented.) irreducible over k, we determine the Brauer group of smooth proper models of X. In a case where it is non-trivial, we exhibit a counterexample to weak approximation.
KW - 14G05 (11D57, 14F22)
UR - http://www.scopus.com/inward/record.url?scp=84925484040&partnerID=8YFLogxK
U2 - 10.1007/s00208-014-1106-7
DO - 10.1007/s00208-014-1106-7
M3 - Article
AN - SCOPUS:84925484040
VL - 361
SP - 1021
EP - 1042
JO - Mathematische Annalen
JF - Mathematische Annalen
SN - 0025-5831
IS - 3-4
ER -